Volume 51, Issue 1B, January 1972

Intensity Distribution of Light upon Reflection from Surface Waves
View Description Hide DescriptionThe actual phase modulation introduced in a light beam by reflection off a surface corrugated in an arbitrary manner is calculated. The forms of the light intensity distribution in the reflection of a uniform‐ or Gaussian‐intensity monochromatic circular light beam from a sinusoidal cylindrically corrugated surface are given.

Acoustic Wave Propagation in Single Crystal
View Description Hide DescriptionAcoustic phase velocity surfaces and ray directions in paratellurite, , have been calculated. Characteristic features of the wave propagation in this highly anisotropic medium have been optically confirmed using the shadow method and Schaefer‐Bergmann diffraction technique. A small misalignment of the wave normal direction has been found to degrade seriously the performance characteristics of acoustic or acousto‐opticaldevices composed of a highly anisotropic medium such as . The existence of the transverse mode with a zero temperature coefficient of velocity has been also confirmed by the present calculation and experiment.

Vibrational Relaxation Times of Methane and Oxygen at Increased Pressure
View Description Hide DescriptionA study of the effect of microphone design on sound absorption and velocity measurements has been carried out for pressures ranging from 1 to 18 atm. Results of this research show clearly that losses may be introduced by the microphone itself and that this problem becomes more severe with increasing pressure. A microphone has been developed which completely eliminates such effects at lower pressures and greatly reduces them at higher pressures.Measurements in gave a pressure dependence of the effective specific‐heat ratio in good agreement with theory. Extrapolation to zero pressure gave a result in good agreement with that obtained by using the vibrational specific heat calculated from the Planck‐Einstein formula. Relaxation times τ_{ v } were determined as a function of pressurep _{0} and lead to the averaged value . This is in good agreement with recent results using the spectrophone method (1.60 μsec⋅atm) and in significant disagreement with results obtained from fluorescence experiments (1.90 μsec⋅atm). Data obtained for gave relaxation times at room temperature and at a temperature of approximately 50°C. The room‐temperature value is higher than previously published data by a factor of 2 and is in reasonably good agreement with extrapolation of high‐temperature shock‐tube data. The value obtained at 50°C is in still better agreement with this extrapolation.

Effect of Particle Viscosity on Propagation of Sound in Suspensions and Emulsions
View Description Hide DescriptionThe effect of particle viscosity, when it is at least one order of magnitude greater than that of the suspending medium, has been included in the formula, existing in the literature, for the velocity of sound in suspensions or emulsions. The corresponding coefficient of viscousabsorption and cross section for scattering of sound energy have also been calculated. The hydrodynamic dispersion of the velocity of sound, due to the frequency‐dependent nature of the viscous interaction between the particles and the suspending medium, has been shown to exist in suspensions or emulsions.

Sound Dissipation by a Small Cylindrical Obstacle
View Description Hide DescriptionScattering of a plane wave, obliquely incident on an infinite‐elastic conducting solid cylinder, which is freely suspended in a viscous conducting fluid, is considered. The wave motion in both fluid and solid is analyzed into viscous (shear), thermal, and dilatational components. Thus the dissipation, associated with the generation of highly damped viscous and thermal fluid waves at the boundary of the cylinder, is calculated. For a given frequency, this is shown to be a function of the cylindrical harmonic amplitudes of the scattered dilatational wave and of the angle of incidence. Results are presented for the simplified situation of normal incidence and small cylinder radius.

Noise Reduction by Vegetation and Ground
View Description Hide DescriptionTransmission of random noise through dense corn, a dense hemlock plantation, an open pine stand, dense hardwood brush, and over cultivated soil was measured. The relation between attenuation and frequency in these diverse cases suggested models that permit the prediction of attenuation in any configuration of vegetation and soil. The corn crop had an excess attenuation of 6 dB/100 ft for each doubling of frequency between 500 and 4000 Hz. On the other hand, the stems of the hemlock, pine, and brush all reduced noise by only about 5 dB/100 ft at 4000 Hz. Bare ground attenuates frequencies of 200–1000 Hz, and the frequency of maximum attenuation depends on the soil permeability to air. Thus, tilling the soil reduced the frequency of peak attenuation from 700 to 350 Hz and increased maximum attenuation at 52 m from the source by nearly 80%. Furthermore, earlier conflicting reports of noise attenuation by vegetation appear reconciled if ground attenuation is taken into account. Scattering and ground attenuation are the principal factors in sound attenuation by vegetation. Both factors attenuate relatively less sound as distance from the sound source increases. Hence measurements far from the source can underestimate the effect of a narrow band of vegetation or soil.

Acoustic Ray Paths through a Model Vortex with a Viscous Core
View Description Hide DescriptionAngular deflection and the formation of a line caustic are examined for acoustic rays passing through a cylindrical vortex with a viscous core. Ray paths are displayed in normalized form and appear to be applicable to the aircraft‐wake problem as well as to long‐distance sound propagation through synoptic weather features.

Reflection‐Refraction of a Stress Wave at a Plane Boundary between Anisotropic Media
View Description Hide DescriptionA review is given of the pertinent equations necessary to describe the reflected and refracted waves at a plane boundary between anisotropic media and the utility of the wave surface in discussing this problem. The critical angle phenomenon in anisotropic media is discussed in terms of the energy flux vector associated with the reflected and refracted modes. The critical angle is shown to occur generally at that angle of incidence for which the energy flux vector of the reflected or refracted mode is parallel to the boundary and not when the wave vector is parallel to the boundary. The possibility of not needing a nonhomogeneous surface wave to satisfy the boundary conditions at angles of incidence greater than the critical angle is discussed for certain particular regions in some anisotropic materials.

Reflection and Transmission of Acoustical Waves from a Layer with Space‐Dependent Velocity
View Description Hide DescriptionThe refraction of acoustical waves by a moving medium layer is theoretically treated and the reflection and transmission coefficients are determined. The moving‐medium‐layer velocity is uniform but with a space dependence in one direction. A partitioning of the moving medium layer into constant‐velocity sublayers is introduced and numerical results for a three‐sublayer approximation of Poiseuille flow are presented. The degenerate case of a single constant‐velocity layer is also treated theoretically and numerically. The numerical results show the reflection and transmission coefficients as functions of the peak moving‐medium‐layer normalized velocity for several angles of incidence.

Vibration and Stability of a Uniformly Curved Tube Conveying Fluid
View Description Hide DescriptionThis paper presents a theoretical study of the vibration and stability of a uniformly curved tube containing flowing fluid. The assumption of the inextensibility of the tube is applied to derive the equation of motion. A solution for the natural frequency is obtained and numerical results are presented. The effects of flow velocity, fluid pressure, and the Coriolis force on the natural frequency are discussed. It is shown that when the flow velocity and fluid pressure exceed a certain value, the tube becomes subject to buckling‐type instability. Critical loads in terms of the flow velocity and fluid pressure are presented for fixed‐fixed, hinged‐hinged, and fixed‐hinged end conditions.

Axisymmetric Vibrations of a Free Finite‐Length Rod
View Description Hide DescriptionThe axisymmetric vibrational characteristics of an elastic circular rod of finite length with stress‐free boundaries are analytically determined, and compared to approximate solutions. The solution approach consists of choosing a series of functions which term by term satisfy the governing equations and the boundary conditions on the shear stress. The boundary conditions on the axial and radial stress are satisfied by an orthogonalization procedure. This method yields an infinite eigenvalue matrix, the coefficients of which are transcendental functions of the frequency. It is found that the procedure converges and that sufficient accuracy is achieved, with truncation to a 20×20 eigenvalue matrix. Comparison is made of these results with the Pochhammer‐Chree solutions which do not permit satisfaction of the boundary conditions on the ends of the rod. This comparison shows many similarities, but also some dissimilarities. Comparisons are also made with the lowest mode of the free‐plate solution and the plane stress solution for very short rods, and with the one‐dimensional rod solution for long rods.

Plane Stress Waves in Membranes Caused by an Arbitrary Pressure Wave
View Description Hide DescriptionUsing the example of the plane motion of a membrane, an analytical method is developed for finding the approximate solution of the second‐order wave equation whose right‐hand side describes a pressure wave moving with decreasing velocity. It is assumed that the variation of the velocity includes the characteristic speed of wave propagation in the membrane. The method is elaborated for obtaining a solution which is valid in the early‐time region. It exploits the Laplace integral transform technique with appropriate approximate methods of inversion. The latter are improved here in comparison with those which have been previously developed in the analyses of similar problems, and can be easily generalized to approximate analyses of the early‐time stage of elastic waves which are caused in thin shells by a pressure wave.

Vibration Attenuation with Beams—Theory and Reciprocal Experiment
View Description Hide DescriptionThe theoretical prediction that significant regions of attenuation exist in the transmissibility curves of cantilever beams driven by dual forces of like magnitude and phase has been substantiated by the results of reciprocal experiments made on small aluminumbeams and a laminated steel/viscoelastic beam. The attenuation provided by the laminated beam, which had high damping, was remarkably extensive and exceeded 15 dB at all frequencies above 100 Hz and 20 dB at all frequencies above 1.2 kHz. Agreement between theoretical prediction and reciprocal experiment has also been noted in two other cases: first, where the large region of attenuation predicted for a uniform cantilever beam was doubled in width and deepened by a factor of more than 10 when the beam was loaded by a suitably positioned lumped mass of similar magnitude to the beam mass (in fact, the resultant region of attenuation was bounded by frequencies that differed in value by a factor of 21 and was greater than 40 dB at its lowest point); and second, where a uniform cantilever beam was driven by forces that were not derived from an isolated source but, rather, were transmitted by two resilient mountings that supported a vibrating mass, which was used to represent a machine. Because the natural frequency of this simple mounting system fell well below the fundamental resonant frequency of the cantilever beam, the mounting system and the beam vibrated more or less independently. In consequence, the attenuation afforded by the simple mounting—essentially 12 dB/octave—superimposed on the region of attenuation noted previously in the beam transmissibility to provide an extremely large and continuous region of attenuation in the transmissibility across the combined simple mounting/beam system at all but low frequencies.

An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder
View Description Hide DescriptionA high‐speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder. Dispersion curves corresponding to real, imaginary, and complex propagation constants for the symmetric and the first four antisymmetric modes of propagation are given. The radial distributions of axial and radial displacements and of shear and normal stresses are given for the symmetric mode. By using a finite number of modes of propagation, an approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface. The reflection coefficient is determined in this way and the accompanying generation of higher order modes at the interface is shown to cause a high‐amplitude end resonance. Experimental results obtained by using the resonance method in conjunction with a long rod are presented to substantiate the calculated reflection coefficient and the frequency of end resonance. Phase velocities, based on measurements of the wavelength of standing waves and resonance frequencies, were obtained for the symmetric and first two antisymmetric modes. These measurements extend into the frequency range of more than one propagating mode. The rms deviation between theoretical and experimental results is in general less than 0.2% with the exception of the dispersion curve for the L(0,2) mode which deviates by 0.7%.

An Extension of the Bessel‐Fubini Series for a Multiple‐Frequency CW Acoustic Source of Finite Amplitude
View Description Hide DescriptionA fundamental operational solution of the lossless Burgers' equation is used to derive the spectrum of a two‐frequency CW source of finite amplitude as a function of range. The resulting expression is then generalized to the case of a multiple N‐frequency CW source of finite amplitude. The region of validity of these solutions is determined by specifying the critical ranges at which the wavefronts become discontinuous. Finally, a modification of these omnidirectional solutions which makes them applicable to directive sources is briefly discussed.

Performance Degradation of Deep‐Ocean Transducers Using Onionskin Paper for Acoustic Decoupling
View Description Hide DescriptionThe acoustic decoupling behavior of onionskin paper is examined, with emphasis on sonartransducer applications. Nonlinear changes in the soundvelocity of onionskin paper with pressure are used to predict the degradation of both frequency and directional response. These results are obtained by computer analysis of a longitudinal‐vibrator equivalent circuit. The results are in reasonable agreement with the measured characteristics of a specific transducer configuration.

Circumferential Waves in a Thin‐Walled Air‐Filled Cylinder in a Water Medium
View Description Hide DescriptionA line source and a line receiver are used to measure values of group and phase velocity for circumferential waves around a thin‐walled air‐filled aluminum cylinder when the cylinder is immersed in water. These are believed to be the first experimental measurements of phase velocity of circumferential waves around a thin‐walled cylinder. Also, experimental values of attenuation are measured for pulses. These experiments are carried out for frequencies from 50 to 200 kHz, corresponding to a value of ka from 15 to 60. A theoreticalanalysis based on the equivalent mechanical impedance of a thin‐walled elastic shell is developed and numerical values are computed for comparison with the experimental data. The phase velocity measurements agree with computed values in their dependence on frequency, but the values are systematically too large by 2%. However, the experimental values measured for pulses do not show good agreement with theory.

Design of Optimum Directional Arrays Using Linear Programming Techniques
View Description Hide DescriptionThe techniques of linear programming have been applied to the synthesis of optimum directional patterns from arrays of point elements. The simplex method has been used to generate optimum element excitations for a given array when the desired directional characteristics are specified. Various criteria are included in the definition of optimality. Numerical examples are given for arrays of up to 70 elements. The size and geometry of arrays that can be treated by this approach are limited only by computation time and computer storage requirements.

Multipath Summability in Ray‐Theory Intensity Calculations in the Real Ocean
View Description Hide DescriptionBy considering the stochastic nature of microfluctuations in the real ocean, the conventional ray‐theory intensity relation is extended to permit consideration of partial coherence in multipath problems. The partial coherence factor is shown to be a function of the curvilinear range, the frequency, and the oceanographic conditions. In addition, a practical estimate for the standard deviation σ_{ I } of the intensity about its expected value is found; σ_{ I } is shown to depend on the partial coherence factor and to saturate as this factor approaches zero.

Normal‐Mode and Ray Theory Applied to Underwater Acoustic Conditions of Extreme Downward Refraction
View Description Hide DescriptionThis theoretical article investigates the acoustic field in the image‐interference and shadow‐zone regions for a model in which the soundvelocity decreases with increasing depth from the surface. Near the surface the shadow‐zone boundary is the surface‐grazing ray, while in deeper regions this boundary is a caustic. The field near the caustic is investigated by modified ray theory based on Airy functions. A comparison of mode theory results with those of ray theory indicates that there is a −π/2 jump in the ray theory phase when the ray touches a caustic. There is, however, no such jump when the ray becomes horizontal. The normal‐mode series is absolutely convergent for ranges greater than 1.732 times the sum of the source and receiver depths. When the shadow zone is bounded by a well‐developed caustic, the series converges so slowly as to be impractical except at ranges well into the shadow‐zone region.